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This paper is concerned with a hybrid reaction–diffusion mosquitoes model perturbed by impulsive control, where the model incorporates imperfect maternal transmission, incomplete cytoplasmic incompatibility (CI) and fitness effect of Wolbachia. The model is periodic and lacks monotonicity due to the integration of impulsive control and imperfect maternal transmission. We establish threshold conditions for the extinction and existence of mosquito populations by the suitable auxiliary problems and lower and upper solutions that can overcome obstacles caused by factors considered, demonstrating the four possible biological outcomes for mosquitoes control. Especially, our theoretical results can reveal the subtle relation between the invasion of Wolbachia and several important parameters including impulsive control rate, impulsive periodic, CI intensity, maternal transmission rate, fitness effect and the initial occupancy of Wolbachia infection. Furthermore, the maximal Wolbachia maternal leakage rate and impulsive control rate against Wolbachia-infected mosquitoes are given. Numerically, we perform simulations to give some interesting phenomena. Specifically, the combination of efficient impulsive control and high proportion of appropriate Wolbachia under certain cases will be more effective in controlling mosquitoes. Surprisingly, the lower initial occupancy, incomplete CI, imperfect maternal transmission and fitness cost of Wolbachia lead to a reversal of Wolbachia invasion from success to failure, but the frequent implementation of impulsive control could again achieve a perfect reversal even for a low initial infection frequency. This work provides new perspectives for further research on mosquitoes control. [ABSTRACT FROM AUTHOR]

In this paper, in order to study comprehensive effect of stage-structure, incomplete immunity and spatial diffusion on the transmission dynamics of sheep brucellosis, we formulate a reaction-diffusion brucellosis model with partial immunity and stage structure in heterogeneous environment. Firstly, the well-posedness of the system is investigated, including the existence of global solution and its ultimate boundedness, and then the basic reproduction number R 0 is defined using the next generation operator. Further, the threshold criteria on the global dynamics of the model are established in terms of R 0 in two special cases. That is, if R 0 < 1 , the disease-free steady state is globally asymptotically stable, while if R 0 > 1 , the model is uniformly persistent and there at least exists a endemic steady state. Furthermore, for the homogeneous space and heterogeneous diffusion model, by constructing suitable Lyapunov functions, we obtain the global asymptotic stability for the disease-free steady-state when R 0 ≤ 1 and the global asymptotic stability endemic steady states when R 0 > 1 . Finally, two simulation examples are given to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the impulsive conformable calculus approach is applied to the introduction of an M 1 oncolytic virotherapy neural network model. The proposed model extends some existing mathematical models that describe the dynamics of the concentrations of normal cells, tumor cells, nutrients, M 1 viruses and cytotoxic T lymphocyte (CTL) cells to the impulsive conformable setting. The conformable concept allows for flexibility in the modeling approach, as well as avoiding the complexity of using classical fractional derivatives. The impulsive generalization supports the application of a suitable impulsive control therapy. Reaction–diffusion terms are also considered. We analyze the stable behavior of sets of states, which extend the investigations of the dynamics of separate equilibrium points. By applying the impulsive conformable Lyapunov function technique, sufficient conditions for the uniform global exponential stability of sets of states are established. An example is also presented to illustrate our results. [ABSTRACT FROM AUTHOR]

Approximate analytical solutions to doubly degenerate reaction-diffusion models pertinent to population dynamics and chemical kinetics have been developed. The double integral-balance method applied to preliminary transformed models and by a direct integration approach has provided physically reasonable results. The model equation scaling has revealed the time and length scales, as well as the characteristic velocity of the process and the Fourier number as the controlling dimensionless group. [ABSTRACT FROM AUTHOR]

Electrodeposition is a low-cost and malleable technique for manufacturing a wide diversity of equipment and materials including coatings and films. It was initially operated to prepare metallic mirrors and corrosion resistant surfaces among other things. We consider a 1D (1 dimensional) stationary problem, that is a mathematical model derived from nickel-iron alloy electrodeposition's chemical reactions.The aim of this paper is to provide a mathematical study of the stationary one dimensional case problem using the topological degree. [ABSTRACT FROM AUTHOR]

Proteins become asymmetrically distributed in the one-cell Caenorhabditis elegans embryo thanks to reaction-diffusion mechanisms that are often entangled in complex feedback loops. Cortical polarity drives the enrichment of the RNA-binding proteins MEX-5 and MEX-6 in the anterior cytoplasm through concentration gradients. MEX-5 and MEX-6 promote the patterning of other cytoplasmic factors, including that of the anteriorly enriched mitotic polo-like kinase PLK-1, but also contribute to proper cortical polarity. The gradient of MEX-5 forms through a differential-diffusion mechanism. How MEX-6 establishes a gradient and how MEX-5 and MEX-6 regulate cortical polarity is not known. Here, we reveal that the two MEX proteins develop concentration asymmetries via similar mechanisms, but despite their strong sequence homology, they differ in terms of how their concentration gradients are regulated. We find that PLK-1 promotes the enrichment of MEX-5 and MEX-6 at the anterior through different circuits: PLK-1 influences the MEX-5 gradient indirectly by regulating cortical polarity while it modulates the formation of the gradient of MEX-6 through its physical interaction with the protein. We thus propose a model in which PLK-1 mediates protein circuitries between MEX-5, MEX-6, and cortical proteins to faithfully establish and maintain polarity. [ABSTRACT FROM AUTHOR]

The two‐step deposition method effectively mitigates the efficiency decline observed in tin‐based perovskite solar cells (TPVSCs) with increasing cell area, stemming from film in‐homogeneity. However, the high solubility of SnI2 in the conventionally used solvent isopropyl alcohol, coupled with the absence of effective modulation of reaction‐diffusion process, results in inadequate film coverage and conversion. In this study, we introduce formic acid as the second‐step solvent and introduce dithiothreitol (DTT) to regulate reaction‐diffusion/crystallization kinetics meticulously. Moreover, this research underscores a fundamental principle that the suitable binding energy ranging from −1.38 to −10.10 kcal mol−1 between ligands and Sn2+ significantly enhances the effectiveness of two‐step crystallization control. Notably, a uniform perovskite film is achieved on large‐scale substrate, and TPVSCs processed with DTT exhibit the highest efficiencies of 12.68 % for 0.04 cm2 device and 11.30 % for 1 cm2 device among tin‐based perovskite devices in two‐step sequential deposition method, even in the absence of dimethyl sulfoxide. This study lays the groundwork for the potential scale‐up development of lead‐free perovskite solar cells. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the initial-boundary value problem of a possibly degenerate reaction-diffusion system over Ω ⊂ R n with n ≥ 1 of the following form ∂ t m i - κ Δ m i + | m i | γ - 2 m i = (∂ x i p) 2 , - ∇ · [ m ∇ p ] = S , with m = diag (m 1 , ⋯ , m n) , the diffusivity κ > 0 , the metabolic exponent γ ≥ 2 and the given function S. When κ = 0 , this system was introduced by Haskovec, Kreusser and Markowich as a continuous version of the discrete Hu-Cai model for biological transport networks. In this work, our result asserts that whenever the random fluctuations of the conductance in the medium were considered, i.e., κ > 0 , then for general large data the corresponding initial-boundary value problem possesses a global weak solution. [ABSTRACT FROM AUTHOR]

Considering the effects of spatial diffusion and environmental noise, this paper establishes a stochastic reaction–diffusion tuberculosis (TB) model that includes varying immunity period. Compared to the classical TB models, the TB models driven by stochastic partial differential equations are more versatile. Mathematically, the boundness of moment on the solutions is given. Furthermore, the regularity of an invariant measure and the unique geometric ergodicity of model are proved by the generalized coupling method. Some numerical simulations are also displayed, which demonstrate the effectiveness of theories. [ABSTRACT FROM AUTHOR]

The aim of this paper was to explore the impact of fear on the dynamics of prey and predator species. Specifically, we investigated a reaction-diffusion predator-prey model in which the prey was subjected to Beddington-DeAngelis type and the predator was subjected to modified Leslie-Gower type. First, we analyzed the existence and stability of equilibria of the nonspatial model, and further investigated the global stability and Hopf bifurcation at the unique positive equilibrium point. For the spatial model, we studied the local and global stability of the unique constant positive steady state solution and captured the existence of Turing instability, which depended on the diffusion rate ratio between the two species. Then, we demonstrated the existence of Hopf bifurcations and discussed the direction and stability of spatially homogeneous and inhomogeneous periodic solutions. Finally, the impact of fear and spatial diffusion on the dynamics of populations were probed by numerical simulations. Results revealed that spatial diffusion and fear both broaden the dynamical properties of this model, facilitating the emergence of periodic solutions and the formation of biodiversity. [ABSTRACT FROM AUTHOR]

In this paper, we study a system of reaction–diffusion equations arising from the competition of two competing species for a single limited nutrient with flocculation in an unstirred chemostat. By the conservation principle, we reduce the dimension of the system by eliminating the equation for the nutrient. Then the global structure of the reduced system is studied by the bifurcation theory in its feasible domain. Finally, we use numerical simulation to verify and supplement our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a reaction–diffusion–advection benthic–drift model, where the population is divided into two interacting groups: individuals dispersing in the drift zone and individuals living in the benthic zone. For different growth types of the benthic population, we present asymptotic profiles of positive steady states in three cases: (i) large advection; (ii) small diffusion of the drift population; and (iii) large diffusion of the drift population. We prove that in case (i) both the benthic and drift individuals concentrate only at the downstream end; in case (ii), both benthic and drift population reside inhomogeneously in (0,L)$(0, L)$, stay away from the upstream end x=0$x = 0$, and concentrate only at the downstream x=L$x = L$; and in case (iii), the drift species distributes evenly on the entire habitat and the benthic species distributes inhomogeneously throughout the habitat. The result supplements the dynamical behaviors of benthic–drift models developed in earlier works and is also of its own interest. [ABSTRACT FROM AUTHOR]

Artificial bio-nanocoatings, primarily composed of proteins, offer a broad range of applications across various fields thanks to their unique properties. Proteins, as major components of these structures, enable a high degree of customization, such as mutations, conjugation with other molecules or nanoparticles, or the inclusion of an enzymatic activity. Their ability to self-assembly simplifies the production of bio-nanocoatings, making this process efficient and environment-friendly. Despite these advantages, a comprehensive understanding of the underlying self-assembly mechanism is lacking, and the reaction rates governing this process have not been characterized. In this article, we introduce a novel method to determine the key parameters describing the self-assembly mechanism of bio-nanostructures. For the first time, this approach enables an accurate calculation of the autocatalytic and self-inhibitory parameters controlling the process. Through mathematical modeling, our method enhances the understanding of how the protein-based nanocoatings form and opens new avenues for their application in nanotechnology and synthetic biology. Improved control over the self-assembly processes may enable the development of nanomaterials optimized for specific functions, such as drug delivery, biosensing, and bioactive surface fabrication. [ABSTRACT FROM AUTHOR]

In this paper we provide general results on the uniform upper semicontinuity of pullback attractors with respect to the time parameter for non-autonomous delay dynamical systems. Namely, we establish a criteria in terms of the multi-index convergence of solutions for the delay system to the non-delay one, locally pointwise convergence and local controllability of pullback attractors. As an application, we prove the upper semicontinuity of pullback attractors for a non-autonomous delay reaction-diffusion equation to the corresponding nondelay one over any bounded time interval as the delay parameter tends to zero. [ABSTRACT FROM AUTHOR]

While reaction-diffusion processes are utilized in multiple scientific fields, these phenomena have seen limited practical application in the polymer industry. Although self-regulating processes driven by parallel reaction and diffusion can lead to patterned structures, most polymeric products with repeating subunits are still prepared by methods that require complex and expensive instrumentation. A notable, high-added-value example is surgical mesh, which is often manufactured by weaving or knitting. In our present work, we demonstrate how the polymer and the biomedical industry can benefit from the pattern-forming capabilities of reaction-diffusion. We would like to propose a self-regulating method that facilitates the creation of surgical meshes from biocompatible polymers. Since the control of the process assumes a thorough understanding of the underlying phenomena, the theoretical background, as well as a mathematical model that can accurately describe the empirical data, is also introduced and explained. Our method offers the benefits of conventional techniques while introducing additional advantages not attainable with them. Most importantly, the method proposed in this paper enables the rapid creation of meshes with an average pore size that can be adjusted easily and tailored to fit the intended area of application. [ABSTRACT FROM AUTHOR]

Asymmetric chemical reactions on the surfaces of colloidal particles are known to propel them into directional motion. The dynamics of such chemical micromotors are sensitive to their local chemical environments, which also continually evolve with the reactions on motor surfaces. This two‐way coupling between the motor dynamics and the local environment may result in complex nonlinear behaviors. As an example, we report that Janus Ag microspheres, which self‐propel in hydrogen peroxide (H2O2), spontaneously reverse their direction of motion two or more times. We hypothesize that two distinct chemical reactions between Ag and H2O2 drive the micromotor in opposite directions, and which reaction dominates depends on the local pH. Interestingly, the local pH near a Ag micromotor oscillates spontaneously in H2O2, likely due to a complex interplay between the kinetics of the reaction between Ag and H2O2 and the diffusion of chemical species. Consequently, the pH‐sensitive Ag micromotor reverses its direction of motion in response to these pH oscillations. This study introduces a new mechanism for regulating the speed and directionality of micromotors, highlights the potential of Ag micromotors in chemical sensing, and sheds new light on the interplay between chemical kinetics and micromotor dynamics. [ABSTRACT FROM AUTHOR]

Mathematical and computational tools have proven to be reliable in decision‐making processes. In recent times, in particular, machine learning‐based methods are becoming increasingly popular as advanced support tools. When dealing with control problems, reinforcement learning has been applied to decision‐making in several applications, most notably in games. The success of these methods in finding solutions to complex problems motivates the exploration of new areas where they can be employed to overcome current difficulties. In this article, we explore the use of automatic control strategies to initial boundary value problems in thermal and disease transport. Specifically, in this work, we adapt an existing reinforcement learning algorithm using a stochastic policy gradient method and we introduce two novel reward functions to drive the flow of the transported field. The new model‐based framework exploits the interactions between a reaction‐diffusion model and the modified agent. The results show that certain controls can be implemented successfully in these applications, although model simplifications had to be assumed. [ABSTRACT FROM AUTHOR]

We study the density fluctuations at equilibrium of the multi-species stirring process, a natural multi-type generalization of the symmetric (partial) exclusion process. In the diffusive scaling limit, the resulting process is a system of infinite-dimensional Ornstein–Uhlenbeck processes that are coupled in the noise terms. This shows that at the level of equilibrium fluctuations the species start to interact, even though at the level of the hydrodynamic limit each species diffuses separately. We consider also a generalization to a multi-species stirring process with a linear reaction term arising from species mutation. The general techniques used in the proof are based on the Dynkin martingale approach, combined with duality for the computation of the covariances. [ABSTRACT FROM AUTHOR]

Asymmetric chemical reactions on the surfaces of colloidal particles are known to propel them into directional motion. The dynamics of such chemical micromotors are sensitive to their local chemical environments, which also continually evolve with the reactions on motor surfaces. This two‐way coupling between the motor dynamics and the local environment may result in complex nonlinear behaviors. As an example, we report that Janus Ag microspheres, which self‐propel in hydrogen peroxide (H2O2), spontaneously reverse their direction of motion two or more times. We hypothesize that two distinct chemical reactions between Ag and H2O2 drive the micromotor in opposite directions, and which reaction dominates depends on the local pH. Interestingly, the local pH near a Ag micromotor oscillates spontaneously in H2O2, likely due to a complex interplay between the kinetics of the reaction between Ag and H2O2 and the diffusion of chemical species. Consequently, the pH‐sensitive Ag micromotor reverses its direction of motion in response to these pH oscillations. This study introduces a new mechanism for regulating the speed and directionality of micromotors, highlights the potential of Ag micromotors in chemical sensing, and sheds new light on the interplay between chemical kinetics and micromotor dynamics. [ABSTRACT FROM AUTHOR]

The transition from inactive to active matter implies a transition from thermodynamically stable to energy‐dissipating structures. Here, we show how the spatiotemporal availability of a chemical fuel causes a thermodynamically stable self‐assembled structure to transiently pass to an energy‐dissipating state. The system relies on the local injection of a weak affinity phosphodiester substrate into an agarose hydrogel containing surfactant‐based structures templated by ATP. Injection of substrate leads to the inclusion of additional surfactant molecules in the assemblies leading to the formation of catalytic hotspots for substrate conversion. After the local disappearance of the substrate as a result of chemical conversion and diffusion the assemblies spontaneously return to the stable state, which can be reactivated upon the injection of a new batch of fuel. The study illustrates how a dissipating self‐assembled system can cope with the intermittent availability of chemical energy without compromising long‐term structural stability. [ABSTRACT FROM AUTHOR]

This paper aims to present a robust computational technique utilizing finite difference schemes for accurately solving time fractional reaction–diffusion models, which are prevalent in chemical and biological phenomena. The time-fractional derivative is treated in the Caputo sense, addressing both linear and nonlinear scenarios. The proposed schemes were rigorously evaluated for stability and convergence. Additionally, the effectiveness of the developed schemes was validated through various linear and nonlinear models, including the Allen–Cahn equation, the KPP–Fisher equation, and the Complex Ginzburg–Landau oscillatory problem. These models were tested in one-, two-, and three-dimensional spaces to investigate the diverse patterns and dynamics that emerge. Comprehensive numerical results were provided, showcasing different cases of the fractional order parameter, highlighting the schemes' versatility and reliability in capturing complex behaviors in fractional reaction–diffusion dynamics. [ABSTRACT FROM AUTHOR]

We consider the effect of a shear flow which has, without loss of generality, a zero mean flow rate, on a Kolmogorov–Petrovskii–Piscounov (KPP)‐type model in the presence of a discontinuous cutoff at concentration u=uc$u = u_c$. In the long‐time limit, a permanent‐form traveling wave solution is established which, for fixed uc>0$u_c>0$, is unique. Its structure and speed of propagation depends on A$A$ (the strength of the flow relative to the propagation speed in the absence of advection) and B$B$ (the square of the front thickness relative to the channel width). We use matched asymptotic expansions to approximate the propagation speed in the three natural cases A→∞$A\rightarrow \infty$, A→0$A\rightarrow 0$, and A=O(1)$A=O(1)$, with particular associated orderings on B$B$, while uc∈(0,1)$u_c\in (0,1)$ remains fixed. In all the cases that we consider, the shear flow enhances the speed of propagation in a manner that is similar to the case without cutoff (uc=0$u_c=0$). We illustrate the theory by evaluating expressions (either directly or through numerical integration) for the particular cases of the plane Couette and Poiseuille flows. [ABSTRACT FROM AUTHOR]

In this paper, we consider reaction-diffusion epidemic models with mass action or standard incidence mechanism and study the impact of limiting population movement on disease transmissions. We set either the dispersal rate of the susceptible or infected people to zero and study the corresponding degenerate reaction-diffusion model. Our main approach to study the global dynamics of these models is to construct delicate Lyapunov functions. Our results show that the consequences of limiting the movement of susceptible or infected people depend on transmission mechanisms, model parameters and population size. [ABSTRACT FROM AUTHOR]

Supramolecular structures are widespread in living system, which are usually spatiotemporally regulated by sophisticated metabolic processes to enable vital biological functions. Inspired by living system, tremendous efforts have been made to realize spatiotemporal control over the self‐assembly of supramolecular materials in synthetic scenario by coupling chemical reaction with molecular self‐assembly process. In this review, we focused on the works related to supramolecular hydrogels that are regulated in space and time using chemical reaction. Firstly, we summarized how spatially controlled self‐assembly of supramolecular hydrogels can be achieved via chemical reaction‐instructed self‐assembly, and the application of such a self‐assembly methodology in biotherapy was discussed as well. Second, we reviewed dynamic supramolecular hydrogels dictated by chemical reaction networks that can evolve their structures and properties against time. Third, we discussed the recent progresses in the control of the self‐assembly of supramolecular hydrogels in both space and time though a reaction‐diffusion‐coupled self‐assembly approach. Finally, we provided a perspective on the further development of spatiotemporally controlled supramolecular hydrogels using chemical reaction in the future. [ABSTRACT FROM AUTHOR]

AbstractResearch in topology optimization with stress constraints has shown that defining the stress constraint locally yields better performance compared to global methods. However, an examination of the formulas for local stress constraints reveals a limitation - the constraint is only satisfied at points where it is explicitly defined, failing to guarantee satisfaction across the entire design domain. To address this shortcoming, this paper proposes utilizing an integral form of the stress constraint. The integral formulation theoretically ensures that the stress constraint is satisfied over whole design domain. The objective is to minimize the mass of plane stress structures using reaction-diffusion level set method while incorporating local stress constraints in this integral form. The methodology utilizes finite element approximations for geometry and displacements, defining local stress constraints through an integral formulation. A Lagrangian function combines objective and constraint functions, with sensitivity analysis performed during optimization based on level set function changes. Structural boundaries are updated using the Hamilton-Jacobi equation. The paper presents numerical examples with varying loads and support conditions to demonstrate the effectiveness of this integral stress constraint approach. Results illustrate the capability of the proposed method to generate optimal topologies while satisfying stress constraints across the entire design space. [ABSTRACT FROM AUTHOR]

Reactive oxygen species (ROS) play a crucial role as signaling molecules in both plant and animal cells, enabling rapid responses to various stimuli. Among the many cellular mechanisms used to generate and transduce ROS signals, ROS-induced-ROS release (RIRR) is emerging as an important pathway involved in the responses of various multicellular and unicellular organisms to environmental stresses. In RIRR, one cellular compartment, organelle, or cell generates or releases ROS, triggering an increased ROS production and release by another compartment, organelle, or cell, thereby giving rise to a fast propagating ROS wave. This RIRR-based signal relay has been demonstrated to facilitate mitochondria-to-mitochondria communication in animal cells and long-distance systemic signaling in plants in response to biotic and abiotic stresses. More recently, it has been discovered that different unicellular microorganism communities also exhibit a RIRR cell-to-cell signaling process triggered by a localized stress treatment. However, the precise mechanism underlying the propagation of the ROS signal among cells within these unicellular communities remained elusive. In this study, we employed a reaction-diffusion model incorporating the RIRR mechanism to analyze the propagation of ROS-mediated signals. By effectively balancing production and scavenging processes, our model successfully reproduces the experimental ROS signal velocities observed in unicellular green algae (Chlamydomonas reinhardtii) colonies grown on agar plates, furthering our understanding of intercellular ROS communication. [Display omitted] [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the positive periodic solutions of the periodic-parabolic problem u t = Δ u + λ u - a (x , t) u p in Ω × (0 , T ] , B u = 0 on ∂ Ω × (0 , T ] , u (x , 0) = u (x , T) in Ω , where Ω is a C 2 + μ bounded domain in R N ( N ≥ 1 ), λ > 0 is a real parameter, p > 1 is constant, a ∈ C μ , μ / 2 (Ω ¯ × [ 0 , T ]) is positive and T-periodic in t. We establish that the positive solution has a "blow-up" phenomenon due to large λ or small a(x, t). By analyzing the sharp profiles, we find that the linear part λ u and nonlinear part a (x , t) u p make quite different effects on the limiting behavior of positive periodic solutions. The second aim is then to investigate the sharp connections between linear and nonlinear parts on the asymptotic behavior of positive periodic solutions. Our study exhibits that the linear part plays a determined role. We also study the asymptotic profiles of periodic-parabolic problem with nonlocal dispersal. We find that the asymptotic profiles are different between two kinds of diffusion problems. [ABSTRACT FROM AUTHOR]

In the realm of discrete-time modeling for gene regulatory networks, significant focus has been placed on addressing the time lags inherent in the process of DNA transcription to RNA and the subsequent translation of mRNA to proteins. These temporal delays have been consistently incorporated into discrete gene regulatory network models. However, true gene regulatory networks are also subject to spatial variables, due to the uneven distribution of protein and mRNA concentrations. The integration of reaction-diffusion terms is thus essential to fully represent the impact of spatial dynamics on gene regulatory networks. In such networks, reaction-diffusion dynamics highlight the complex interactions between neighboring spatial regions, where closeness leads to mutual influences on their functional activities. Based on this conceptual groundwork, this study introduces a discrete-time gene regulatory network model that includes the mutual interconnections between spatial areas. To guarantee the model's robust stability, we have established delay-dependent stability criteria using carefully designed Lyapunov-Krasovskii functions, framed within the context of linear matrix inequalities. The robustness and effectiveness of our approach are demonstrated through a numerical example presented in this work. [ABSTRACT FROM AUTHOR]

We study the reaction diffusion equation $ u_t-du_{xx} = f(u) $ with a monostable nonlinear function $ f(u) $ over a changing interval $ [g(t), h(t)] $, viewed as a model for the spreading of a species with population range $ [g (t), h(t)] $ and density $ u(t,x) $. The free boundaries $ x = g(t) $ and $ x = h(t) $ are not governed by the same Stefan condition as in Du and Lin [20] and other previous works; instead, they satisfy a related but different set of equations obtained from a 'preferred population density' assumption at the range boundary, which allows the population range to shrink. We obtain a rather complete understanding of the longtime dynamics of the model, which exhibits persistent propagation with a finite asymptotic propagation speed determined by a certain semi-wave solution, and the density function converges to the semi-wave profile as time goes to infinity. The asymptotic propagation speed is always smaller than that of the corresponding classical Cauchy problem where the reaction-diffusion equation is satisfied for $ x $ over the entire real line with no free boundary. Moreover, when the preferred population density used in the free boundary condition converges to 0, the solution $ u $ of our free boundary problem converges to the solution of the corresponding classical Cauchy problem, and the propagation speed also converges to that of the Cauchy problem. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the study of adaptive finite-time synchronization (FTS) for generalized delayed fractional-order reaction-diffusion quaternion-valued neural networks (GDFORDQVNN). Utilizing the suitable Lyapunov functional, Green's formula, and inequalities skills, testable algebraic criteria for ensuring the FTS of GDFORDQVNN are established on the basis of two adaptive controllers. Moreover, the numerical examples validate that the obtained results are feasible. Furthermore, they are also verified in image encryption as the application. [ABSTRACT FROM AUTHOR]

In this research, a novel method for regulating components in RE2SiO5/RE2Si2O7 multiphase silicates was developed, combining the benefits of a suitable thermal expansion coefficient (CTE) and outstanding corrosion resistance against calcium–magnesium–alumino–silicate (CMAS). This approach enhanced the overall thermophysical properties. Additionally, the results from the CMAS corrosion resistance test indicated that (Lu1/3Yb1/3Tm1/3)2SiO5/(Lu1/3Yb1/3Tm1/3)2Si2O7 and (Lu1/4Yb1/4Tm1/4Er1/4)2SiO5/(Lu1/4Yb1/4Tm1/4Er1/4)2Si2O7 exhibited exceptional resistance to CMAS penetration, even at temperatures up to 1500 °C. To comprehend the corrosion mechanism of CMAS on these silicates, we introduced a reaction–diffusion model, which involved observing the changes in the interface between the corrosion product layer and the silicate block. This was achieved using electron backscatter diffraction (EBSD). These findings lay a theoretical basis for selecting rare earth elements in RE2SiO5/RE2Si2O7 multiphase silicates based on the radii of different rare earth cations.

This paper presents a brucellosis disease model with reaction–diffusion and time delay. The model takes into account both the direct and indirect transmission of infected animals and pathogens in the environment. By analyzing the associated characteristic equation, the local stability of the unique positive equilibrium point is established. The existence of Hopf bifurcations at the positive equilibrium point is also examined by considering the discrete time delay as a bifurcation parameter. Additionally, an optimal control analysis is conducted to minimize disease outbreaks and control costs. This includes reducing the exposure of susceptible animals to infected animals, removing infected animals from herds, and reducing emissions of brucella into the environment. By constructing Hamiltonian function and applying Pontryagin’s maximum principle, the necessary conditions for the existence of optimal control are given. Finally, the existence of bifurcation periodic solutions and the effectiveness of control strategies are illustrated through numerical simulations. [ABSTRACT FROM AUTHOR]

To explore the efficacy of integrated pest management, we modify the predator–prey (pest–natural enemy) model by incorporating Holling III functional response and transform it into a non-smooth Filippov control system. Unlike conventional Filippov systems, the model takes into consideration time delay and spatial heterogeneity. Consequently, we establish and examine a delayed reaction–diffusion Filippov prey–predator model. Firstly, the dynamics of the two subsystems are analyzed, which includes the existence and stability of the equilibrium points, along with determining the adequate conditions for local Hopf bifurcation. Subsequently, we implement a detailed investigation of the sliding mode dynamics and stability of the pseudoequilibrium. Theoretical and numerical simulations indicate that on the one hand, the threshold level should be prescribed adequately to reduce the pest population equal to or below the threshold level. On the other hand, reading from the boundary node and boundary focus bifurcations, slightly varying the economic threshold may save a failure control strategy by dragging the number of the pests from a regular equilibrium above the threshold to a boundary equilibrium or a pseudoequilibrium equal to the threshold. Furthermore, the sequent appearance of global sliding bifurcations including touching, sliding switching and crossing bifurcations expound that the incorporation of time delay not only complicates the dynamics of the system, but also brings more challenge for pest control. [ABSTRACT FROM AUTHOR]

A class of time-varying delay impulsive reaction–diffusion tree–grass–water–nitrogen system driven by Lévy jump process is considered. First, we prove the existence and uniqueness of the global positive solution of the model by constructing the Lyapunov function. Secondly, several sufficient conditions for finite-time stability are given by using comparison theorem and mean impulse interval method. Finally, numerical simulations are carried out to verify the effectiveness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

A singularly perturbed reaction–diffusion problem in 1D is solved numerically by a local discontinuous Galerkin (LDG) finite element method. For this type of problem the standard energy norm is too weak to capture the contribution of the boundary layer component of the true solution, so balanced norms have been used by many authors to give more satisfactory error bounds for solutions computed using various types of finite element method. But for the LDG method, up to now no optimal-order balanced-norm error estimate has been derived. In this paper, we consider an LDG method with central numerical flux on a Shishkin mesh. Using the superconvergence property of the local L 2 projector and some local coupled projections around the two transition points of the mesh, we prove an optimal-order balanced-norm error estimate for the computed solution; that is, when piecewise polynomials of degree k are used on a Shishkin mesh with N mesh intervals, in the balanced norm we establish O ((N - 1 ln N) k + 1) convergence when k is even and O ((N - 1 ln N) k) when k is odd. Numerical experiments confirm the sharpness of these error bounds. [ABSTRACT FROM AUTHOR]

During the outbreak of major emerging infectious diseases, virus droplets can survive in the environment as aerosols for hours to days, and their impact on human infectious diseases is often overlooked. In addition, the speed of transmission of infectious diseases is often closely related to transportation. Therefore, studying the impact of environmental viruses and transportation on disease development is significant for effective infectious disease prevention and control. We proposed a degenerate reaction–diffusion infectious disease model (SEAIR) considering environmental virus interference and established a well-posedness and threshold system for this model. We have obtained the system solution approaches the disease-free equilibrium (피0) when the basic reproduction number ℛ0 ≤ 1. The system has at least one positive steady state solution (PSS) when ℛ0 > 1. This paper used a non-standard finite difference method discretization model while data fitting and parameter estimation were performed based on data provided by the Health Commission. Further sensitivity analysis was conducted on ℛ0. At the same time, we also discussed the impact of various parameters in the early stages of the outbreak of major emerging infectious diseases on the development of the disease. Research found that even if the contact rate between people is controlled at a shallow level, the disease may persist. In the early stages of major emerging outbreak of infectious diseases, immediately reducing the use of transportation can effectively reduce the speed of disease spread. [ABSTRACT FROM AUTHOR]